The continuous and changing impact of affect on risky decision-making

Affective experience has an important role in decision-making with recent theories suggesting a modulatory role of affect in ongoing subjective value computations. However, it is unclear how varying expectations and uncertainty dynamically influence affective experience and how dynamic representation of affect modulates risky choices. Using hierarchical Bayesian modeling on data from a risky choice task (N = 101), we find that the temporal integration of recently encountered choice parameters (expected value, uncertainty, and prediction errors) shapes affective experience and impacts subsequent choice behavior. Specifically, self-reported arousal prior to choice was associated with increased loss aversion, risk aversion, and choice consistency. Taken together, these findings provide clear behavioral evidence for continuous affective modulation of subjective value computations during risky decision-making.


Affective experience model
The momentary valence and arousal responses were modeled using the exponential decay of the affective influence of previously encountered events. The models included terms for expected value (EV), uncertainty (U), and prediction error (PE; see main text for computation of the terms). We tested alternative models including models with outcomes instead of EV and PE, without uncertainty, or with different parametrizations of PE. Model 1 contained separate terms for gamble EV, and PE. γi is a forgetting factor adjusting the influence of recent events in comparison to earlier events with 0 ≤ γ ≤ 1. Hence, model 1, including an individual variance parameter (see Model Parametrization below), contains 5 parameters per participant.
We fit an alternative model containing a term for the gamble U to verify that the gamble uncertainty influences affective experience (model 2), which improved the model fit (see Table S1).
To verify that affective experience depends on expectations and prediction errors rather than actual outcomes, we next fit a model containing a term for the outcome received on each trial instead of expectations and prediction errors (model 3). Model 3 indicated that expectations and prediction errors explain more variance in affective experience than actual outcomes (Table S1). Additionally, we fit two additional models (variations of model 2) that included different parametrizations of PE, which both improved the model fit compared to model 2 (Table S1). Model 4 included separate terms for positive and negative PE to capture the differential influence of positive and negative surprise on valence and arousal (see ). Model 5 included signed and unsigned PE terms, which captures the impact of the magnitude of PE independent of its direction (Equation S4). This model was included to investigate whether affective experience (especially the arousal feature of affect) is more sensitive to the size of PE rather than its direction. The model fit of these five different models indicated that model 5 provided the best fit for both valence and arousal ratings (Table S2), which suggests that momentary affective experience reflects temporal integration of expectations, uncertainty, and the direction and magnitude of prediction errors.

Model parametrization and fit
Individual-level parameters were drawn from group-level normal distributions.
Additionally, we used a non-centered parametrization to speed up estimation (Ahn et al., 2017;Stan Development Team, 2018). All the weight parameters, including the constant term, were parameterized as follows: Here, ́ represents individual-level variations from the group mean ( ), and w in Equation S5 is mathematically equivalent to ~ ( , ).
The forgetting factor (γ) was parametrized similarly with only difference being the probit transformation (the cumulative distribution function of a unit normal distribution) of the individual level parameters to ensure 0 ≤ γ ≤ 1.
The observed valence and arousal ratings in the experiment were modeled using a normal distribution, ( , , ). , is the individual specific location changing in every trial and calculated according to the model (Equation S4). is the individual standard deviation sampled as follows (here the exponential transformation ensures a positive value for the individual level variance).

Parameter recovery
We modeled the choice behavior based on subjective utility computations with two parameters: loss aversion (λ) and risk sensitivity (ρ). The softmax choice rule with an inverse temperature parameter (i.e. choice consistency; c) was used to compute the probability of accepting the gamble (Equations 5-7 in the main manuscript). We used hierarchical Bayesian analysis to estimate group and individual level parameters (see below for Model Parametrization and Fit). Since we used a novel risky choice task, we carried out a parameter recovery analysis to ensure that the study and the modeling procedure can provide reliable results under ideal conditions.
We simulated 100 datasets (each with 101 subjects) randomly selecting group level parameters according to the following: This selection ensured a wide range of group level parameters, which were used to select individual level parameters (N=10100 in total) and generate individual data. Then, the choice model was fit to each dataset using HBA to estimate both group and individual level parameters. Figure S3 shows the correlations between simulated and estimated parameters.
The results show that group level means for ρ (Pearson-R = .998) and λ (Pearson-R = .999) were recovered reliably. The choice consistency parameter was recovered reliably within 0 to 2 range. However, around 30% of simulations, in which c was higher than 2, were overestimated (Pearson-R = .67). The individual level parameters for ρ (Pearson-r = .988) and λ (Pearson-r = .996) were also recovered reliably. We also found that 95% HDIs of the posterior distribution of parameters contained the original parameter values generating the dataset in 93% of the simulations. In addition, Table S2 summarizes correlations between estimated parameters, which indicates that ρ and λ was recovered independently. Taken together, these simulations indicate that loss aversion and risk sensitivity parameters could be recovered independently over a reasonable range under ideal conditions, while the choice consistency could be recovered reliably within the range from 0 to 2.  Next, we investigated the parameter combinations to identify the cases in which µc is overestimated. We found that the combination of µ  > 1 (which implies increasing marginal utility) and µc > 2 resulted in the overestimation of µc ( Figure S4). To verify this finding, we simulated 1000 data points with several combinations of µ and µc (with µ=1). Figure S5 shows the expected value of the gambles against the probability of accepting for various data generating parameters. The simulations verify that when the risk sensitivity parameter is large different choice consistency levels do not differentiate. Taken together, these findings suggest that the current task design and analysis is limited when the data generating process is defined by increasing marginal utility and high choice consistency.

Model parametrization and fit
We used a non-centered parametrization and individual-level parameters were drawn from group-level normal distributions. Loss aversion was parameterized as follows:   In addition, we fit two models in which the effect of experienced arousal was tested while controlling for the previous outcome (control model 1) and previous PE (control model 2). In these models, both experienced arousal and either the previous outcome or the previous PE are allowed to modulate the decision parameters. The results showed that the modulatory effect of arousal on all decision parameters persisted when controlling for previous gamble outcome and PE. While the influence of the gamble outcome and PE from the previous trial was not different from zero.